Recall that a graph is called an \(\alpha \) -vertex expander if there exists \(\alpha \in (0,1)\)  such that every subset \(S\)  of vertices with at most \(|V|/2\)  vertices has at least \(\alpha |S|\)  neighbors not in \(S\) . Previous approaches for proving the existence of desired graphs as minors in expanders [1]}, [2]} obtain an embeddingSee Subsection REF  for further details on minor-theoretic concepts we use. of a graph \(H\)  as a minor in \(G\)  where every connected subgraph of \(G\)  corresponding to a vertex in \(H\)  is of size \(O(\log n)\) . To achieve nearly tight lower bounds we need every connected subgraph to be of size \(O(\log n/ \log d)\) . We obtain this stronger upper bound by adapting the approach of Krivelevich and Nenadov [2]} (which considers vertex expanders) to give the desired bounds for graphs that have sufficiently strong edge expansion properties.
